find the first and the 15 ters of an arithmetic sequence whose fifth term is -1 and ninth term is 11.
The difference between the consecutive terms is 3.
fifth term = a + 4d = -1
ninth term = a + 8d = 11
subtract them:
4d = 12
d = 3, as Ms. Sue stated
plug back into a + 4d = -1
a + 12 = -1
a = -13 <--- first temr
now use your formulas to find the 15th term.
To find the first and 15th terms of an arithmetic sequence, we need to first find the common difference (d) between the terms.
The formula for any term in an arithmetic sequence is given by:
a(n) = a(1) + (n - 1)d
Where:
- a(n) represents the nth term
- a(1) represents the first term
- n represents the term number
- d represents the common difference
Given that the fifth term is -1 and the ninth term is 11, we can plug the values into the formula to create a system of equations:
-1 = a(1) + (5 - 1)d
11 = a(1) + (9 - 1)d
Simplifying these equations will give us the values of a(1) and d, which will allow us to find the first and 15th terms.
-1 = a(1) + 4d --(1)
11 = a(1) + 8d --(2)
Now, let's solve this system of equations to find the values of a(1) and d:
Subtract equation (1) from equation (2):
11 - (-1) = (a(1) + 8d) - (a(1) + 4d)
12 = 4d
Now, divide both sides by 4:
d = 3
Now, substitute the value of d into either equation (1) or (2) to find a(1):
-1 = a(1) + 4(3)
-1 = a(1) + 12
Subtract 12 from both sides:
-13 = a(1)
Therefore, the first term (a(1)) of the arithmetic sequence is -13, and the common difference (d) is 3.
To find the first term (a(1)) and the 15th term (a(15)), we can use the formula:
a(n) = a(1) + (n - 1)d
For the first term (n = 1):
a(1) = -13 + (1 - 1) * 3
a(1) = -13
For the 15th term (n = 15):
a(15) = -13 + (15 - 1) * 3
a(15) = -13 + 14 * 3
a(15) = -13 + 42
a(15) = 29
Therefore, the first term of the arithmetic sequence is -13, and the 15th term is 29.